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108 V NORMED SPACES

The definition of the norm in JSf (E; F) depends on the norms in E and
in F; but it is readily seen that, when the norms in E and F are replaced
by equivalent norms (Section 5.6) the new norm in <£ (E; F) is equivalent
to the old one.

(5.7.5) Let u be a continuous linear mapping of a normed space E into a
normed space F, and v a continuous linear mapping ofF into a normed space G.

Then \\v*u^ \\v\\ -\\u\\.

For if |W| ^ 1, then by (5.7.4) \\v(u(x))\\ < \\v\\ - \\u(x)\\ ^ \\v\\ - ||i/||,
and the result follows from (5.7.1).

(5.7.6) If F is a real (resp. complex) normed space, the mapping which to
each aeF associates the element 9_a: £ -*%aof & (R; F) (resp. J5?(C; F)) is
a linear isometry off onto JSf (R; F) (resp. J?(C; F)).

The mapping a-+0_a is obviously linear; it is surjective, for every linear
mapping/of R (resp. C) into F is such that/(£) =/(£ • 1) = £/(!) = & with
a =/(!). Finally ||0J = sup \\fr\\ = \\a \\ by axiom (III) of Section 5.1.

LetnowE_1? ..., E_n,Fbera H- I normed spaces, and define Sf(E_i9 . . ., E_M;F)
as the vector space of all continuous multilinear mappings of E_i x • • • x E_ninto F. Then for u e J2?(E_l9 . . . , E_M; F), the same argument as above shows
that the g.Lb. \\u\\ of all constants a > 0 such that

||«(^₁,....,x_ll)||<fl||x₁||---|Wl

is also given by

(5.7.7) l|_tt||= sup Kx_l5...₅x_n)||.

ll*i||«l,...,||*n||«l

We also see that \\u\\ is a norm on ££ (E_l3 . . . , E_n ; F) ; but in fact these vector
spaces can be reduced to spaces j? (X ; Y) :

(5.7.8) For each u e & (E, F; G) and each x e E, let u_x be the linear mapping
y -> u(x, y). Then u: x -> u_x is a linear continuous mapping of E into ££ (F; G),
and the mapping u-*iiisa linear isometry of<£ (E, F ; G) onto J?(E ; jSf (F ; G)).

We have \\u_x(y)\\ = llw^^)!! '< \\^u\\ '\\^x\\ ' \\y\\* hence u_x is continuous
by (5.5.1); moreover ||w_x|| = sup \\u(x, y)\\, hence (2.3.7)

|= sup



	108 V NORMED SPACES The definition of the norm in JSf (E; F) depends on the norms in E and in F; but it is readily seen that, when the norms in E and F are replaced by equivalent norms (Section 5.6) the new norm in <£ (E; F) is equivalent to the old one. (5.7.5) Let u be a continuous linear mapping of a normed space E into a normed space F, and v a continuous linear mapping ofF into a normed space G. Then \\vu^ \\v\\ -\\u\\.* For if \|W\| ^ 1, then by (5.7.4) \\v(u(x))\\ < \\v\\ - \\u(x)\\ ^ \\v\\ - \|\|i/\|\|, and the result follows from (5.7.1). (5.7.6) If F is a real (resp. complex) normed space, the mapping which to each aeF associates the element 9_a: £ -%aof &* (R; F) (resp. J5?(C; F)) is a linear isometry off onto JSf (R; F) (resp. J?(C; F)). The mapping a-+0_a is obviously linear; it is surjective, for every linear mapping/of R (resp. C) into F is such that/(£) =/(£ • 1) = £/(!) = & with a =/(!). Finally \|\|0J = sup \\fr\\ = \\a \\ by axiom (III) of Section 5.1.

	LetnowE_1? ..., E_n,Fbera H- I normed spaces, and define Sf(E_i9 . . ., E_M;F) as the vector space of all continuous multilinear mappings of E_i x • • • x E_ninto F. Then for u e J2?(E_l9 . . . , E_M; F), the same argument as above shows that the g.Lb. \\u\\ of all constants a > 0 such that \|\|«(^₁,....,x_ll)\|\|<fl\|\|x₁\|\|---\|Wl is also given by (5.7.7) l\|_tt\|\|= sup Kx_l5...₅x_n)\|\|. lli\|\|«l,...,\|\|n\|\|«l We also see that \\u\\ is a norm on ££ (E_l3 . . . , E_n ; F) ; but in fact these vector spaces can be reduced to spaces j? (X ; Y) :

	(5.7.8) For each u e & (E, F; G) and each x e E, let u_x be the linear mapping y -> u(x, y). Then u: x -> u_x is a linear continuous mapping of E into ££ (F; G), and the mapping u-iiisa linear isometry of<£* (E, F ; G) onto J?(E ; jSf (F ; G)).

	We have \\u_x(y)\\ = llw^^)!! '< \\^u\\ '\\^x\\ ' \\y\\* hence u_x is continuous by (5.5.1); moreover \|\|w_x\|\| = sup \\u(x, y)\\, hence (2.3.7)

	\|= sup